Answer :
To determine the reliability and validity of the thermometers based on the given temperature readings, we will follow a step-by-step process:
1. Calculate Reliability (Standard Deviation):
Reliability is assessed by calculating the standard deviation of the measurements for each thermometer. The standard deviation measures the spread of the data points around the mean and indicates how consistent the readings are.
The formula for the standard deviation, [tex]\(\sigma\)[/tex], is:
[tex]\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \][/tex]
where [tex]\(N\)[/tex] is the number of observations, [tex]\(x_i\)[/tex] is each individual observation, and [tex]\(\mu\)[/tex] is the mean of the observations.
- Thermometer W:
Readings: [tex]\(100.1, 99.9, 96.9\)[/tex]
- Mean ([tex]\(\mu\)[/tex]) = [tex]\(\frac{100.1 + 99.9 + 96.9}{3} = 98.9667\)[/tex]
- Standard deviation ([tex]\(\sigma_W\)[/tex]) ≈ 1.4636
- Thermometer X:
Readings: [tex]\(100.4, 102.3, 101.4\)[/tex]
- Mean ([tex]\(\mu\)[/tex]) = [tex]\(\frac{100.4 + 102.3 + 101.4}{3} = 101.3667\)[/tex]
- Standard deviation ([tex]\(\sigma_X\)[/tex]) ≈ 0.7760
- Thermometer Y:
Readings: [tex]\(90.0, 95.2, 98.6\)[/tex]
- Mean ([tex]\(\mu\)[/tex]) = [tex]\(\frac{90.0 + 95.2 + 98.6}{3} = 94.6\)[/tex]
- Standard deviation ([tex]\(\sigma_Y\)[/tex]) ≈ 3.5365
- Thermometer Z:
Readings: [tex]\(90.8, 90.6, 90.7\)[/tex]
- Mean ([tex]\(\mu\)[/tex]) = [tex]\(\frac{90.8 + 90.6 + 90.7}{3} = 90.7\)[/tex]
- Standard deviation ([tex]\(\sigma_Z\)[/tex]) ≈ 0.0816
The most reliable thermometer will have the lowest standard deviation. Based on these calculations, Thermometer Z, with a standard deviation of approximately 0.0816, is the most reliable.
2. Calculate Validity (Closeness to 100.0°C):
Validity is assessed by calculating the absolute difference between the mean of the readings and the actual boiling point of pure water (100.0°C).
- Thermometer W:
- Mean ([tex]\(\mu_W\)[/tex]) = 98.9667
- Validity (|difference|) = |100.0 - 98.9667| ≈ 1.0333
- Thermometer X:
- Mean ([tex]\(\mu_X\)[/tex]) = 101.3667
- Validity (|difference|) = |100.0 - 101.3667| ≈ 1.3667
- Thermometer Y:
- Mean ([tex]\(\mu_Y\)[/tex]) = 94.6
- Validity (|difference|) = |100.0 - 94.6| ≈ 5.4000
- Thermometer Z:
- Mean ([tex]\(\mu_Z\)[/tex]) = 90.7
- Validity (|difference|) = |100.0 - 90.7| ≈ 9.3000
The thermometer with the highest validity will have the smallest absolute difference from 100.0°C. Based on these calculations, Thermometer W, with a difference of approximately 1.0333, has the highest validity.
### Conclusion
Based on the data:
- A. Thermometer Z is the most reliable.
- D. Thermometer W has the highest validity.
1. Calculate Reliability (Standard Deviation):
Reliability is assessed by calculating the standard deviation of the measurements for each thermometer. The standard deviation measures the spread of the data points around the mean and indicates how consistent the readings are.
The formula for the standard deviation, [tex]\(\sigma\)[/tex], is:
[tex]\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \][/tex]
where [tex]\(N\)[/tex] is the number of observations, [tex]\(x_i\)[/tex] is each individual observation, and [tex]\(\mu\)[/tex] is the mean of the observations.
- Thermometer W:
Readings: [tex]\(100.1, 99.9, 96.9\)[/tex]
- Mean ([tex]\(\mu\)[/tex]) = [tex]\(\frac{100.1 + 99.9 + 96.9}{3} = 98.9667\)[/tex]
- Standard deviation ([tex]\(\sigma_W\)[/tex]) ≈ 1.4636
- Thermometer X:
Readings: [tex]\(100.4, 102.3, 101.4\)[/tex]
- Mean ([tex]\(\mu\)[/tex]) = [tex]\(\frac{100.4 + 102.3 + 101.4}{3} = 101.3667\)[/tex]
- Standard deviation ([tex]\(\sigma_X\)[/tex]) ≈ 0.7760
- Thermometer Y:
Readings: [tex]\(90.0, 95.2, 98.6\)[/tex]
- Mean ([tex]\(\mu\)[/tex]) = [tex]\(\frac{90.0 + 95.2 + 98.6}{3} = 94.6\)[/tex]
- Standard deviation ([tex]\(\sigma_Y\)[/tex]) ≈ 3.5365
- Thermometer Z:
Readings: [tex]\(90.8, 90.6, 90.7\)[/tex]
- Mean ([tex]\(\mu\)[/tex]) = [tex]\(\frac{90.8 + 90.6 + 90.7}{3} = 90.7\)[/tex]
- Standard deviation ([tex]\(\sigma_Z\)[/tex]) ≈ 0.0816
The most reliable thermometer will have the lowest standard deviation. Based on these calculations, Thermometer Z, with a standard deviation of approximately 0.0816, is the most reliable.
2. Calculate Validity (Closeness to 100.0°C):
Validity is assessed by calculating the absolute difference between the mean of the readings and the actual boiling point of pure water (100.0°C).
- Thermometer W:
- Mean ([tex]\(\mu_W\)[/tex]) = 98.9667
- Validity (|difference|) = |100.0 - 98.9667| ≈ 1.0333
- Thermometer X:
- Mean ([tex]\(\mu_X\)[/tex]) = 101.3667
- Validity (|difference|) = |100.0 - 101.3667| ≈ 1.3667
- Thermometer Y:
- Mean ([tex]\(\mu_Y\)[/tex]) = 94.6
- Validity (|difference|) = |100.0 - 94.6| ≈ 5.4000
- Thermometer Z:
- Mean ([tex]\(\mu_Z\)[/tex]) = 90.7
- Validity (|difference|) = |100.0 - 90.7| ≈ 9.3000
The thermometer with the highest validity will have the smallest absolute difference from 100.0°C. Based on these calculations, Thermometer W, with a difference of approximately 1.0333, has the highest validity.
### Conclusion
Based on the data:
- A. Thermometer Z is the most reliable.
- D. Thermometer W has the highest validity.
